differential calculus

Decomposing multi-factor causal responsibility in complex systems—such as legal reasoning, AI decision-making, biological pathways, and climate models—remains fundamentally challenging due to entangled causal interactions. Method: This paper introduces the first intervention-based causal effect decomposition framework, integrating insights from partial information decomposition (PID) with Möbius inversion on a redundancy lattice. It constructs a closed-form Möbius function enabling exact quantification—under genuine causal interventions—of synergistic, redundant, and unique causal effects. Unlike correlation-based approaches, the framework supports context-sensitive causal power analysis. Results: Empirically validated on logic gates, cellular automata, and chemical reaction networks, the method reveals systematic evolutionary patterns of causal components with respect to structural topology and parametric variation. It provides both theoretical foundations and computational tools for AI explainability, biological pathway dissection, and other domains requiring rigorous causal attribution.

Existing causal models struggle to distinguish between the immediate and persistent effects of interventions in time-dynamic systems, particularly when such interventions alter the system’s equilibrium behavior. This work proposes a novel paradigm grounded in system and state representations, integrating causal directed acyclic graphs, the potential outcomes framework, and dynamic systems theory. By introducing an equilibrium-state assumption and employing state-space modeling, the study reformulates the causal inference framework to better capture temporal dynamics. It innovatively defines an equilibrium-oriented “zero effect” concept and combines it with a strategic selection of time points to enable valid identification of time-varying causal parameters. The approach establishes clear criteria for categorizing causal effects under dynamic interventions, substantially enhancing the interpretability and practical utility of causal inference in equilibrium analysis.

This work addresses the problem of jointly identifying dynamical laws and causal structure from observed time-series data generated by ordinary differential equation (ODE)-driven systems, while enabling counterfactual prediction under interventions. We propose a novel neural ODE framework that—uniquely—integrates lightweight sparsity and symmetry regularization to achieve robust dynamics modeling and causal graph learning even under non-identifiable conditions. The method unifies the representation of inter-variable dynamics and causal dependencies, supporting explicit intervention inference on both variables and system parameters. Evaluated across diverse synthetic benchmarks—including linear and nonlinear first- and second-order ODE systems—as well as real-world datasets, our approach significantly improves dynamical reconstruction accuracy and counterfactual prediction reliability, while enhancing causal interpretability through structured, sparse, and symmetric Jacobian estimation.

This work addresses fundamental challenges in scientific modeling—namely, parameter unidentifiability in dynamical systems, out-of-distribution (OOD) inference, and difficulty in estimating intervention effects—by establishing, for the first time, a rigorous theoretical connection between causal representation learning and dynamical systems theory. Methodologically, it integrates identifiability constraints, neural ordinary differential equations (Neural ODEs), differentiable ODE solvers, and structured latent-variable modeling to jointly ensure trajectory-level parameter identifiability and model scalability. The resulting framework learns controllable, interpretable dynamical representations from high-dimensional, entangled observational data, enabling downstream causal tasks such as OOD classification and treatment effect estimation. Experiments on a synthetic wind-field simulator and real-world climate datasets demonstrate that the model accurately quantifies the impact of external forcings (e.g., greenhouse gas emissions) on temperature trends—yielding estimates consistent with established climate science consensus.

This paper addresses the problem of identifying total causal effects from observational time series under realistic constraints where only abstract causal graphs—either extended summary graphs (preserving lagged/instantaneous causal structure) or summary graphs (fully discarding temporal information)—are available. We establish, for the first time, a theoretical identifiability framework for total effects under such abstract time-series causal graphs: we prove that total effects are always identifiable under extended summary graphs; for summary graphs, we derive sufficient conditions for identifiability and provide a principled criterion for constructing valid adjustment sets. Furthermore, we propose a general, consistent adjustment-set construction method that supports interpretable and computationally feasible causal effect estimation in high-dimensional settings and in the presence of latent confounders. Our work provides a novel theoretical foundation for time-series causal inference under limited causal prior knowledge.

Traditional causal discovery methods in dynamic systems are often constrained by assumptions of acyclicity or stationarity, rendering them ill-suited for real-world complexities such as feedback loops, cyclic interactions, and nonstationarity. This work proposes a hybrid causal discovery framework that integrates physical knowledge with data-driven learning by embedding known physical mechanisms as inductive biases into a stochastic differential equation (SDE) model: the drift term encodes established ordinary differential equation (ODE) dynamics, while the diffusion term captures unknown causal couplings. By systematically incorporating partial physical priors into dynamic causal discovery for the first time, the approach overcomes conventional limitations and substantially enhances the identifiability and robustness of the inferred causal graph. Coupled with sparsity-inducing maximum likelihood estimation, the proposed algorithm outperforms state-of-the-art purely data-driven baselines across multiple dynamic system benchmarks, yielding more accurate, stable, and physically consistent causal structures.

This work addresses the challenge of causal discovery in dynamic systems where delayed or overlapping causal effects render traditional observational methods ineffective. Focusing on chain-reaction systems characterized by cascading activations, the authors propose a causal identification strategy based on blocking interventions: by selectively preventing component activation, the true causal structure can be uniquely determined with only a small number of targeted interventions. Theoretical analysis demonstrates that the proposed method achieves exponential error decay and logarithmic sample complexity under finite-sample conditions. Empirical evaluations on both synthetic models and diverse chain-reaction environments confirm its efficacy, substantially outperforming purely observational heuristic approaches.

This study addresses the challenge of structural unidentifiability and inference difficulty in nonlinear dynamic systems operating on unknown interaction networks. The authors propose an identification framework based on an implicit dependence matrix, establishing necessary and sufficient conditions for network identifiability by revealing its reliance on the spectral heterogeneity of the interaction matrix. The framework characterizes observational equivalence classes and overcomes the limitation of conventional approaches that erroneously conflate network dependencies with common shocks. Methodologically, it integrates semiparametric estimation, spectral analysis, and asymptotic theory to construct estimators with desirable asymptotic properties and develops a test for network dependence whose power is governed by spectral characteristics. The proposed framework demonstrates broad applicability across economic systems, including production networks and contagion models.

Existing graphical causal models struggle to represent the causal structure of continuous-time dynamical systems governed by stochastic differential equations (SDEs), particularly lacking theoretical foundations regarding interventional distributions, marginalization, and sample-path properties. This work establishes a rigorous semantic framework for causal SDEs, introducing solvability conditions that ensure well-defined observational and interventional distributions under marginalization. It formally develops σ-separation-based Markov properties and do-calculus rules within the SDE setting for the first time. For additive-noise SDEs, the paper proves that a stronger d-separation property holds even in cyclic systems. A time-rescaling transformation is introduced to unify subsampling, Granger non-causality, and local independence. These results demonstrate that constraint-based causal discovery algorithms—such as PC, FCI, CCD, and CCI—can be directly applied in this continuous-time framework.

This work addresses the problem of determining whether the sign of the drift coefficient associated with a specific edge in the causal graph of a continuous-time linear stationary stochastic differential equation is identifiable when the diffusion matrix is unknown. The authors introduce the notion of “edge sign identifiability” and develop a unified analytical framework that does not require prior knowledge of the diffusion matrix, thereby accommodating cyclic structures and classical causal models such as instrumental variables. By integrating causal graphical models with covariance matrix analysis, they derive general graphical criteria for sign identifiability—fully identifiable, non-identifiable, or partially identifiable—and provide explicit expressions for the target edge’s sign in terms of observed covariances for several canonical graph structures. This approach substantially relaxes the common assumption of known diffusion terms and broadens the applicability of sign inference for causal effects in continuous-time settings.